A Compendium on Linearisation in Continuum Kinematics

Abstract This work presents the linearisation, about an arbitrary motion, of the main kinematical quantities in Solid Mechanics and, as a novel result, of the polar decomposition of the deformation. When the polar decomposition is linearised about the identity motion (undeformed configuration), one retrieves the classical result that the multiplicative polar decomposition of the deformation gradient becomes the additive decomposition of the displacement gradient additive. The setting is in affine spaces, but the covariant framework allows for a relatively straightforward extension to Riemannian manifolds.